It is well known that when two parts presenting relatively high coefficients of thermal expansion are in intimate contact, and one of the elements in contact is at a temperature higher than ambient temperature, then heat will propagate to the other part, thereby causing said other part to expand thermally.
When the part is of dimensions that are large compared with the contacting surfaces that constitute the heating points, the temperature at each point along the part varies and becomes distributed so as to present isothermals in the material constituting the part around the heating points. This situation may be unstable over time if the transmission of energy through the contact surface(s) is not constant, if ambient temperature is not constant, or if the heat dissipated through the set of contacting surfaces is less than that transmitted by the heating surface(s).
When free thermal expansion is not possible, thermal tensions in compression and deformation become present locally at the site of the local heating, and the magnitudes thereof can be calculated.
Regardless of whether the thermal expansion is free or involves internal thermal tensions, the result is mechanical deformation of the part.
Given the shape of the part and the locations of the heating points, thermal tensions can give rise overall to maximum deformations that are considerably greater than would happen if the part were heated uniformly. This situation is created by the fact that even though the local heating surface areas are very small, they create local thermal tensions and deformations that are strong and that deform zones remote from the heating surfaces by a “lever arm” system, even when those zones are themselves subjected to little heating. It is precisely the temperature difference between the various portions of a single part that make free thermal expansion impossible without creating internal thermal tension.
The inventors have shown that this situation is to be found particularly, but not exclusively, in controlling the displacement of mechanical members in machine tools, and in particular when controlling press brakes for folding sheet metal.
FIG. 1 is a diagram of a press brake 10 with a frame 12, a moving top die 14 that carries fastener members 16 for securing to folding tools, and a bottom die 18 that carries fastener members for folding V-shapes 20. In conventional manner, movement in translation of the top die 14 is controlled by two hydraulic rams 22 and 24 having cylinders 22a, 24a that are secured to the frame 12 and that have the ends of their pistons 22b, 24b secured to the top ends of the moving die 14. The dimensions of the top die are of the order of:                length: 1200 millimeters (mm) to 6000 mm;        height: 1000 mm to 3000 mm; and        thickness: 40 mm to 120 mm.        
Tests carried out by the inventors have shown that the hydraulic rams 22 and 24, like most other force generators, such as electromechanical ball-screw actuators, have a continuous operating temperature that exceeds ambient temperature by a few degrees or a few tens of degrees Celsius. That heating is produced by various internal sources of friction between moving and stationary parts, by hysteresis in the material that is subjected to reciprocating loading by hot hydraulic oil passing into the ram, and continuously heating the piston. The hydraulic oil passing through the ram can reach 70° C. and the piston can reach about 60° C. The faces of the pistons 22b and 24b of the rams close to the action surface 26 on the top die 14 are at about 40° C. while ambient temperature is 17° C., and the central portions of the top die are at about 20° C. As can be seen more clearly in FIG. 2, these temperatures going away from the contact surface 26 between the ram 24 and the top die 14 are distributed and present isothermals I. When a press brake, only the deformation of the bottom edge 28 of the top die, as represented by line J, is significant. For the performance of the press brake, it is this deformation that is significant and it results from the superposition of the two above-described phenomena:                firstly, linear thermal expansion is free and there is no thermal stress in the vertical direction. A mean temperature rise of 10° in this zone that has a height of 400 mm gives a theoretical linear expansion of 48 micrometers (μm) in application of a well-known formula, which deformation corresponds to that which can be measured on a real machine; and        secondly, the deformation of the central portion of the top die is produced by thermal tensions associated with the heating of the material beneath and beside the action points of the rams.        
In present press brakes, the positions of the control rams can be defined with accuracy of 1 μm to 2 μm. It is therefore unacceptable for deformation of the bottom edge 28 of the top die 14 being capable of reaching 48 μm.